I am having a hard time getting my head around Functionals and Calculus of Variations,

My question is: Given a functional and using the Euler-Lagrange equation to find an extremal,

The question I am working on is

\(\displaystyle J(y) = \int_{0}^{1} ((y')^2 -y)dx\) with \(\displaystyle y(0)=0, y(1)=1\)

I found the extremal to be: \(\displaystyle y(x) = \frac{-1}{4}x^2 +\frac{5}{4}x\) which I am told is a minimum to the functional problem.

However I am unsure on what is sufficient to show this, in the notes I have it is shown that:

\(\displaystyle J(y+f) = J(y) + \int_{0}^{1}(f')^2dx \geq J(y)\) where f is continuously differentiable on the interval 0,1 with \(\displaystyle y(0)=y(1)=0\)

Thanks in advance!

My question is: Given a functional and using the Euler-Lagrange equation to find an extremal,

*how do we show that*the extremal provides a min/max (if it does)The question I am working on is

\(\displaystyle J(y) = \int_{0}^{1} ((y')^2 -y)dx\) with \(\displaystyle y(0)=0, y(1)=1\)

I found the extremal to be: \(\displaystyle y(x) = \frac{-1}{4}x^2 +\frac{5}{4}x\) which I am told is a minimum to the functional problem.

However I am unsure on what is sufficient to show this, in the notes I have it is shown that:

\(\displaystyle J(y+f) = J(y) + \int_{0}^{1}(f')^2dx \geq J(y)\) where f is continuously differentiable on the interval 0,1 with \(\displaystyle y(0)=y(1)=0\)

Thanks in advance!

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